1. Field of the Invention
This invention relates to a generator for generating an orthogonal sequence the components of which take various kinds of complex numbers and to a radar system including such an orthogonal sequence generator.
2. Prior Art
Before describing the prior art, the mathematical properties of the orthogonal sequences will first be described.
The term "sequence" used herein means time series of numerical values a.sub.n shown in expression 1, below: EQU {a.sub.n }=. . . a.sub.n-1 a.sub.n a.sub.n+1 . . . (1)
n is a factor representing the order of the sequences. a.sub.n is referred to as a component of this sequence and is a complex number. The sequence {a.sub.n } is a periodic sequence in which there exists an integral number of N satisfying expression 2, below: EQU a.sub.n+N =a.sub.n ( 2)
Thus, the sequence of expression 1 can be denoted by expression 3, below: EQU {a.sub.n }=. . . a.sub.N-1 a.sub.0 a.sub.1 a.sub.2 . . . a.sub.N-1 a.sub.0 a.sub.1 . . . (3)
In order to provide a quantum for describing the mathematical properties of such a sequence, an autocorrelation function as defined by expression 4, below, is often used: ##EQU2## wherein, * represents a complex conjugate. The reason why the autocorrelation function is defined only within the range from m=0 to m=N-1, as shown in expression 4, is that the sequence {a.sub.n } is a periodic series and thus the autocorrelation function .rho.(m) is a periodic function. The period thereof is N and the same as that of the sequence {a.sub.n }. Thus the function .rho.(m) satisfies the following expression: EQU .rho.(m+N)=.rho.(m) (5)
When such a sequence is applied to a practical system, it is necessary that the autocorrelation function of expression 4 has such properties as shown in FIG. 1 of U.S. Pat. No. 4,939,745, infra i.e., the function of autocorrelation has a sharp peak at m=0 and takes a "considerably low value" in the remaining range of m(m=1, . . . , N-1). The portion .rho.(0) of the function at m=0 is referred to as the main lobe, and the other portion .rho.(m) (m=1, . . . , N-1) of the function than that at m=0 is referred to as the side lobe, and the magnitude of the side lobe relative to the main lobe .rho.(0) poses a problem to be discussed. The magnitude of the side lobe which is a "considerably low value" must thus satisfy relation 6, below: EQU .vertline..rho.(m).vertline.&lt;&lt;.vertline..rho.(0).vertline. (6) EQU (m=1,2, . . . ,N-1)
With the satisfaction of relation 6, the sequence having zero magnitude of the side lobe of the autocorrelation function, i.e., satisfying expression 7, below, has excellent properties: ##EQU3## The orthogonal sequence is defined as satisfying expression 7.
Among the systems for generating orthogonal sequences having such properties, there are complex two-value orthogonal sequence generating systems as disclosed, for example, in U.S. Pat. No. 4,939,745 by the same inventors as those of this application and entitled "Apparatus for Generating Orthogonal Sequences" and polyphase orthogonal sequence generating systems as disclosed in the article of "Phase Shift Pulse Codes with Good Periodic Correlation Properties" by R. Frank, et al., IRE Transaction Information Theory, Vol. IT-8, pp. 381-382; October 1962, U.S.A.
FIG. 1 shows an arrangement of the orthogonal sequence generator disclosed in U.S. Pat. No. 4,939,745. The generator comprises a two-element M-sequence generator on GF(2) 3 consisting of a linear feedback shift-resistor and a component substituting device 4 having substitution means for substituting a component a.sub.n for a component b.sub.n of a two-element M-sequence {b.sub.n } output from the device 4 and outputting it. FIG. 2 is a flowchart showing the operation of the component substituting device 4 of FIG. 1. The above-mentioned substituting device sets the values of the component a.sub.n to the following complex number, corresponding to the component b.sub.n being 0 or 1: EQU a.sub.n =A.sub.o exp(j.phi..sub.0)=A.sub.0 e.sup.j.phi..sbsp.o (when b.sub.n =0) (8) EQU a.sub.n =A.sub.1 exp(j.phi..sub.1)=A.sub.1 e.sup.j.phi..sbsp.1 (when b.sub.n =1) (9)
A.sub.o, .phi..sub.o, A.sub.1 and .phi..sub.1 are set to satisfy the following expression: ##EQU4## where N is a period of the M-sequence. Since A.sub.o =1 and .phi..sub.o =0, normally, expression 10 is denoted by the following expression: ##EQU5##
The sequence {a.sub.n } is indicated as an orthogonal sequence and the period N is the same as that of the M-sequence {b.sub.n } which is denoted by N=2.sup.k -1. The component a.sub.n of the sequence {a.sub.n } takes two complex number values and thus the physical quantity one-to-one corresponding thereto is limited to two kinds.
FIG. 3 is a vector diagram of the component a.sub.n of the orthogonal sequence {a.sub.n }. The component a.sub.n can take two values, 1 and Ae.sup.j.phi..
FIG. 4 shows a flowchart of creating the polyphase orthogonal sequence described in the above-referred IRE Information Theory. This polyphase orthogonal sequence {a.sub.n } has, as components, the L-power root of 1 (w) denoted by expression 12 and the power of L-power root of 1 (w.sup.k) denoted by expression 13: EQU w=exp (j2.pi./L)(L is an integer.gtoreq.2) (12) EQU w.sup.k =exp (j2.pi.k/L)(k is an integer) (13)
Thus, the sequence can be denoted by the following expression: ##EQU6##
The period N of the polyphase orthogonal sequence {a.sub.n } is uniquely determined by L and denoted as N=L.sup.2. As shown with expression 13, the amplitude of the component w.sup.k of the sequence {a.sub.n } does not depend on the value of k, but 1, and its phase takes an L-number of values from 0 to 2.pi.(L-1)/L at every interval of (2.pi./L). Moreover, once the period N has been determined, the sequence is determined into only one form.
FIG. 5 is a vector diagram showing an example component group {w.sup.k } of the polyphase orthogonal sequence {a.sub.n } at L=8. In this case, the amplitude of all the components are 1 and the phase takes 8 values at every .pi./4.
Turning now to a radar system provided with an orthogonal sequence generator, there is a radar system provided with a complex two-value orthogonal sequence, as disclosed, for example, in U.S. Pat. No. 4,939,745 mentioned above.
FIG. 6 shows an arrangement of a radar system disclosed in the above U.S. patent. As shown in the drawing, it comprises a local oscillator 11 for generating a sinusoidal wave signal e.sup.j.omega.t, an orthogonal sequence generator 13 arranged as shown in FIG. 1 for generating a complex two-value orthogonal sequence {a.sub.n }, a modulator 12 for code-modulating the sinusoidal wave signal e.sup.j.omega.t by using the complex two-value orthogonal sequence {a.sub.n }, a power amplifier 14 for amplifying a code-modulated transmission signal U and radiating an output through a transmission antenna 16 to an external space, a low-noise amplifier 15 for amplifying a reception signal R received by a reception antenna 17 including reflection signals Sa and Sb from two targets 10a and 10b and for transferring the amplified signal, a detector 18 for converting the reception signal R in the RF band to a detection signal V in the video band and a demodulator 19 for performing the correlation process operation of the detection signal V and the sequence {a.sub.n } to output a demodulated signal Z.
The basic operation of the radar system shown in FIG. 6 will now be described.
In order to simplify the description, the mathematical expression of the signals will be made by complex signals. As shown by the Euler's formula of the following expression 15, the real signal can correspond to the real part of the complex signal: EQU e.sup.j.omega.t =exp (j.omega.t)=cos .omega.t+j sin .omega.t(15)
where, j: imaginary unit.
FIGS. 7A-7D are diagrams showing timing relationships between the transmission and reception signals U and R shown in FIG. 6. In FIGS. 7A-7D, the code-modulated transmission signal U, reflected signal Sa from the target 10a, reflected signal Sb from the target 10b and reception signal R are respectively illustrated in explanatory form.
As shown in the drawing, the change-over of the component a.sub.n is performed at every period of time .tau. so that a component a.sub.o is used in the time interval between t=0 and t=.tau. and a component a.sub.1 is used in the time interval between t=.tau. and t=2.tau., . . . , thereby code-modulating the sinusoidal wave signal e.sup.j.omega.t generated by the local oscillator 11 to provide the transmission signal U.
The code-modulated transmission signal U is denoted by the following expression: ##EQU7## where, rect(t) is a rectangular function as defined by the following expression: ##EQU8##
The modulation is expressed by the product of the component a.sub.n of the sequence {a.sub.n } and the sinusoidal wave signal e.sup.j.omega.t. Since the sequence {a.sub.n } is a periodic series, the modulated transmission signal U is also a periodic sequence having a period of T=N.tau..
Because the reflected signals Sa and Sb are created by reflecting a part of the transmission signal U on the targets, the waveforms thereof are similar to the waveform of the transmission signal U. However, the timing each of the reflected signals Sa and Sb being received by the reception antenna 17 is delayed by such a time as required for the radio wave to propagate twice the slant range between the radar system and each target. In FIGS. 7B and 7C, such time delays are indicated by ta and tb with respect to the reflected signals Sa and Sb, respectively. Thus, the mathematical expressions Sa(t), Sb(t) of the reflected signals Sa, Sb are expressed by the following expressions: ##EQU9## where, .eta..sub.a, .eta..sub.b are constant values representing the intensities of reflection of the radio waves on the targets 10a and 10b.
Since the reception signal R is a compound signal including both the reflected signals Sa and Sb, its mathematical expression R(t) is given as follows: ##EQU10##
The detector 18 phase-detects the signal R and this can be expressed as the multiplication of the signal R(t) and exp(-j.omega.t). Thus, the detection signal V can be expressed as follows: ##EQU11##
The correlation process of the detection signal V(=V.sub.(t)) and the sequence {a.sub.n } is performed in the demodulator 19. The detection signal V output from the detector 18 is sampled in the demodulator 19 and then converted to a digital signal. The sampling period in this case is set to be the same as the time period .tau. of changingover the components of the sequence shown in FIG. 7A. The detection signal V(k.tau.) (k= . . . , -1, 0, 1, . . . ) which is converted to the digital signal can be expressed as follows: ##EQU12## where, t.sub.a =k.sub.a .multidot..tau., and t.sub.b =k.sub.b .multidot..tau..
Taking into consideration that the rectangular function rect(t) is 0 out of range of 0.ltoreq.t&lt;1, as shown in expression 16b, expression 20 can be simply expressed as follows: EQU V(k)=.eta..sub.a exp(-j.omega..tau.k.sub.a)a.sub.k-k.sbsb.a +.eta..sub.b exp(-j.omega..tau.k.sub.b)a.sub.k-k.sbsb.b ( 21)
The demodulator 19 performs a correlation process operation as shown in the following expression using the sampled and converted detection signal V (=V.sub.(k.tau.)) and the complex two-value orthogonal sequence {a.sub.n } provided from the orthogonal sequence generator 13 to output a demodulated signal Z (=Z.sub.(k)): ##EQU13##
Expression 22 is substituted by the following expression by using expression 21: ##EQU14##
Comparing expression 23 with expression 4, the terms parenthesized by [ ] in expression 23 represent autocorrelation functions of the complex two-value orthogonal sequence {a.sub.n }, and thus expression 23 can be rewritten using the autocorrelation function .rho.(m) to obtain the expression, below: EQU Z(k)=.eta..sub.a exp(-j.omega..tau.k.sub.a).rho.(k-k.sub.a)+.eta..sub.b exp(-j.omega..tau.k.sub.b).rho.(k-k.sub.b) (24)
As shown by expression 24, the demodulated signal Z(k) is in the form of adding the autocorrelation functions of the sequences regarding the signals Sa and Sb.
FIGS. 8A-8C show waveforms of the amplitude of the demodulated signal Z(k). FIG. 8C shows amplitude-waveforms in the case of the sequence being orthogonal, while FIGS. 8A and 8B show amplitude-waveforms in the case of the sequence being non-orthogonal.
The radar system shown in FIG. 6 can derive such advantages that since the sequence {a.sub.n } employed in the code-modulation operation is an orthogonal sequence having the side lobe of the autocorrelation function of 0, even when there is a substantial difference between the radiowave reflection intensities .eta..sub.a and .eta..sub.b on the adjacent two targets, the two-target signals Za and Zb can be detected from the demodulated signal Z(k) without the main lobe of the signal Zb having a small-magnitude being covered by any side lobe Ya of the large-magnitude signal Za as shown in FIG. 8C.
On the contrary, the radar system using the complex two-value orthogonal sequence for code modulation has a property such that when the period N of the sequence is relatively large and if any other electronic device which receives such a transmission signal includes a square detector, an output of the square detector would be a sinusoidal wave signal, whereby the angular frequency .omega. of the transmission signal could easily be detected.
That is, in the case of the period N of the sequence being large, expression 11 can approximately be expressed as follows: ##EQU15## Since cos .phi..ltoreq.1, A=1 and .phi.=.pi. are obtained from expression 25.
Then, the code-modulated transmission signal U(t) can be approximately expressed by the following expression: ##EQU16## where, .phi..sub.n =0 when a.sub.n =1, and .phi..sub.n =.pi. when a.sub.n =-1.
When expressing the U(t) in the form of real signal, it is expressed as follows: ##EQU17##
In the case of such a transmission signal U(t) being received by any other electronic device which is provided with a square detector, an output signal Y(t) of the square detector can be expressed as follows: ##EQU18## Since 2.phi..sub.n is 0 or 2.pi., irrespective of the value n, Y(t) can take the following relation: EQU Y(t)=(cos 2.omega.t+1)/2 (29)
Thus, Y(t) indicates a sinusoidal wave signal, and if the frequency components of the signal Y(t) is analyzed by a spectrum analyzer, the angular frequency .omega. of the transmission signal U(t) will be able to be simply detected, even when the complex two-value orthogonal sequence {a.sub.n } is unknown.
FIG. 9 shows an exemplary arrangement of the modulator 12 shown in FIG. 6 in the event that the already-described complex two-value orthogonal sequence {a.sub.n } is used for code modulation. The components a.sub.n of the sequence {a.sub.n } takes two complex number values and thus the physical quantity corresponding one-by-one thereto is limited to two kinds. Accordingly, the number of change-over of the phase at the modulator is two irrespective of the period of the sequence and, as a consequence, the required number of phase shifters is one.
FIG. 10 also shows another exemplary arrangement of the modulator 12 shown in FIG. 6 in the event that the already-described conventional polyphase orthogonal sequence w.sup.k is used for code modulation. In the case that the period N of the polyphase orthogonal sequence is 64, the number L of change-over of the phase at the modulator is 8 (=.sqroot.N), and thus the required number of phase shifters is 7 (=.sqroot.N-1).
In the above-mentioned prior generators for generating an orthogonal sequence having a property that the side lobe of the autocorrelation function thereof is 0, the components thereof take two complex numbers and thus the physical quantities corresponding one-to-one thereto one limited to only two kinds. On the contrary, the polyphase orthogonal sequences are limited to one sequence existing with respect to a particular period N. However, in the case of a multi-value sequence signal being required, or a plurality of sequence signals with respect to the particular sequence period N being required in order to improve interference-resisting and cross talk-preventing performances of a radar system, a communication system, a home automation system or a factory automation system, there has been a problem as to have to produce such kinds of orthogonal sequence.
In a radar system, moreover, a purpose of providing a code modulation using a sequence is to convert a transmission signal to a quasi-noise signal and then radiate it to an external space so that in the transmitted radio signal is hardly detected by any other electronic device, for example other radars. In such a kind of radar system, it is particularly important that the angular frequency .omega. of the transmission signal cannot be detected by any other electronic device. However, there has been a problem such that when the prior complex two-value orthogonal sequence is employed for code, modulation, if the period thereof is relatively large, the angular frequency .omega. of the transmission signal from the radar system can easily be detected by any other electronic device which receives such a transmission signal.
Further, when the conventional polyphase orthogonal sequence is employed for code modulation in a radar system, the number of change-overs of the phase at a modulator depends on the period of the polyphase orthogonal sequence, and thus if the period of the sequence is large, the arrangement of the modulator becomes complicated.